Indice degli argomenti
General presentation
This course is divided into two parts: Prof. Julien Melleray from the University of Lyon 1 held the first part on Topological Dynamics, and Prof. Eduardo Duéñez from the University of Texas at San Antonio will hold the second part on Functional Analysis.
The second part will begin on May 2, 16:30-18:30.
The course will be taught completely in English.
Introduction to Topological Dynamics
Here is what we covered:- First lecture: Introduction to Polish groups, discussion of some examples and first applications of Baire category methods (e.g. the Pettis lemma). Roughly p 1-6 of the lecture notes.
- Second lecture: end of the proof of Theorem 1.14, quotients of Polish groups (p7 and first half of p8). Then discussion of category-preserving maps and the Kuratowski-Ulam theorem (From def 1.22 at the bottom of p11 to the end of the first chapter). The proof of Theorem 1.28 was not given on the board.
- Third lecture: discussion of orbits for Polish group actions, Rosendal's criterion for existence of a comeager orbit, Effros theorem; we more or less covered all of chapter 1.
- Fourth lecture: Quick introduction to first order structures, ultrahomogeneity and Fraïssé theory. We covered chapter 2 except for the end, about the extension property.
- Fifth lecture: equivalence between extension property and existence of an increasing chain of compact subgroups with dense union (for the automorphism group of a Fraïssé limit in a relational language). Introduction of uniform structures, caracterisation of Hausdorfness and a metrizability criterion. Existence and uniqueness of a compatible uniform structure on a compact Hausdorff space.
- Sixth lecture: filters as a way to study topology, a brief discussion. Definition of a Cauchy filter and a discussion of complete uniform spaces and the notion of a Hausdorff completion (without proofs). The four natural uniform structures on a topological group; proof that they are all compatible with the group topology. Birkhoff-Kakutani theorem: every Hausdorff, first-countable topological group is metrizable by a left-invariant metric. Proof that the left-completion of a Polish group is compact iff the group is compact.
- Seventh lecture: introduction of Roelcke precompact groups, proof the oligomorphic groups are Roelcke precompact, isometric actions of Roelcke precompact groups have bounded orbits. Discussion of the Stone--Cech compactification of the integers, general notion of a compactification.
- Eight lecture: Hausdorff compactifications of topological spaces; Gelfand-Naimark theorem relating Hausdorff compactifications and closed, unital, *-subalgebras of the algebra of continuous bounded functions. Definition of the Samuel compactification of a uniform space and discussion of the role of right-uniformly continuous bounded functions.
- Ninth lecture: Proof that (S(G),1) is the greatest ambit. Definition of a universal minimal flow M(G) and proof that it exists. Right-topological semigroup structure on S(G), Ellis lemma about existence of idempotents and proof of the uniqueness of M(G) up to isomorphism as a consequence of its coalescence. Definition of extreme amenability.
- Tenth lecture: Identification of a class of flows that can be used as test spaces for extreme amenability for nonarchimedean Polish groups. Ramsey property for a Fraïssé class, equivalence between its finite and infinite version (badly done on the board).
- Eleventh lecture: we discussed again the equivalence between the finite and infinite versions of the Ramsey property, and proved that this property is equivalent to extreme amenability. As an example, we showed that the class of linear orders has the Ramsey property, by proving the classical Ramsey theorem. Then we discussed the notion of a co-precompact subgroup, and connected the action of S_infinity/Aut(Q) to the action of S_infinity on the compact space of all linear orders (a compact, minimal flow).
- Twelth lecture: Description of the universal minimal flow of S_infinity, discussion of a general strategy to compute metrizable universal minimal flows and proof that a metrizable universal minimal flow of a Polish group must have a comeager orbit. Brief discussion of Polish metric structures and their automorphism groups.
Below are the full lecture notes; we covered every part but sometimes skipped some details/proofs. It is certain that there are still some typos, hopefully none major...Functional Analysis
This 24-hour (12-lecture) course will focus on structures with real-valued predicates, with primary emphasis on vector spaces (such as normed vector spaces and lattices, Banach spaces, Lₚ-spaces and dual pairs). The underlying framework is that of Keisler's "general structures" with real-valued predicates.
Note: The Notes cover background material (on basic analysis, topology, etc.) that will not necessarily be covered in lectures, but is there for reference. Since the course content will adapt to the background of students, the Notes (below) are expected to change rapidly. Please re-download the Notes file each time you access this page! (Grazie!)
- Lecture 1: Types in metric spaces.
- Lecture 2: Real-valued structures.
- Lecture 3: Normed spaces, Banach spaces, Banach duals and Alaoglu's Theorem.
- Lecture 4: The Hahn Banach and Open Mapping Theorems.
- Lecture 5: 𝓁ₚ spaces and the Stone Representation Theorem.
- Lecture 6: Banach vector lattice-algebras.
- Lecture 7: L1,∞ pairs. Representation as classical measure spaces à la Riesz.
- Lecture 8: First-order real-valued languages.
- Lecture 9: Ultraproducts. Łos’s Theorem. Compactness of first-order real-valued logic.
- Lecture 10: Stable formulas and definable predicates in real-valued logic. Radon-Nikodym Theorem in L1,∞-pairs.
- Lecture 11: Proof of stability of L1. Statement of related stability results (Lₚ- and Orlicz spaces). Survey of “representability” results (Krivine and Krivine-Maurey).
- Lecture 12: Omitting Types Theorem in real-valued logic.